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The problems are generally based on the following model: A particular spacecraft can be represented as a single axisymmetric rigid body B. Let n₂ be inertially fixed unit vectors; then, 6, are parallel to central, principal axes. To make the mathematics simpler, introduce a frame C where n₂ = ĉ₁ = b; initially. 6₁ Assume a mass distribution such that J =₁₁• B* •b₁ = 450 kg - m² I = b² •Ï¾˜ • b₂ = b¸ •Ï¾* •b¸ = 200 kg - m² K J-I C³ =r₁₁ = r₁₁ Assume that the body moves in a circular orbit at a constant rate Q. Let â₁ be orbit-fixed unit vectors where a₂ is directed from the orbit center to B*; then â³ is 90° from a₂ in the direction of motion and â₁ is parallel to the orbit normal. ده * The dependent variables in the differential equations that govern motion include the measure numbers ₁; the kinematic variables that are the body 1-3-2 angles for orientation of C in A: @₁ = 0 w₂ = -Kw₁₂-rw₁₂+3KQ² (c₁₂S¸ − c₁₁ ) (c₁₁₁ + §¸§₁ ) w₁ = Kw₁₂+rw₂ −3KQ² (c₁S₂c₁₂ + $₁₂S₁) c₁₂ Ġ₁ = [(@₁c¸ + @¸§¸ − rc³)/2] - Ω ė₁₂ = − (@₁ − 1 ) s³ + W₂cz - - Ò¸ = {[@¸¢¸ − r¢¸+@₂$¸ ] $₂ / €₂ } + @₂ $2 The motion of interest is a constant spin of B in N about an axis parallel to the orbit normal, i.e., ³ = @â₁ = nÒâ₁, for some given n. Select 7 appropriately to keep C fixed in A during the nominal motion. If r = x2, what is the proper choice of x? Why? What is the resulting particular solution of the differential equations that corresponds to the nominal motion? Demonstrate that it satisfies the differential equations.

The problems are generally based on the following model: A particular spacecraft can be represented as a single axisymmetric rigid body B. Let n₂ be inertially fixed unit vectors; then, 6, are parallel to central, principal axes. To make the mathematics simpler, introduce a frame C where n₂ = ĉ₁ = b; initially. 6₁ Assume a mass distribution such that J =₁₁• B* •b₁ = 450 kg - m² I = b² •Ï¾˜ • b₂ = b¸ •Ï¾* •b¸ = 200 kg - m² K J-I C³ =r₁₁ = r₁₁ Assume that the body moves in a circular orbit at a constant rate Q. Let â₁ be orbit-fixed unit vectors where a₂ is directed from the orbit center to B*; then â³ is 90° from a₂ in the direction of motion and â₁ is parallel to the orbit normal. ده * The dependent variables in the differential equations that govern motion include the measure numbers ₁; the kinematic variables that are the body 1-3-2 angles for orientation of C in A: @₁ = 0 w₂ = -Kw₁₂-rw₁₂+3KQ² (c₁₂S¸ − c₁₁ ) (c₁₁₁ + §¸§₁ ) w₁ = Kw₁₂+rw₂ −3KQ² (c₁S₂c₁₂ + $₁₂S₁) c₁₂ Ġ₁ = [(@₁c¸ + @¸§¸ − rc³)/2] - Ω ė₁₂ = − (@₁ − 1 ) s³ + W₂cz - - Ò¸ = {[@¸¢¸ − r¢¸+@₂$¸ ] $₂ / €₂ } + @₂ $2 The motion of interest is a constant spin of B in N about an axis parallel to the orbit normal, i.e., ³ = @â₁ = nÒâ₁, for some given n. Select 7 appropriately to keep C fixed in A during the nominal motion. If r = x2, what is the proper choice of x? Why? What is the resulting particular solution of the differential equations that corresponds to the nominal motion? Demonstrate that it satisfies the differential equations.

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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The problems are generally based on the following model: A particular spacecraft can be represented as a single axisymmetric rigid body B. Let n₂ be inertially fixed unit vectors; then, 6, are parallel to central, principal axes. To make the mathematics simpler, introduce a frame C where n₂ = ĉ₁ = b; initially. 6₁ Assume a mass distribution such that J =₁₁• B* •b₁ = 450 kg - m² I = b² •Ï¾˜ • b₂ = b¸ •Ï¾* •b¸ = 200 kg - m² K J-I C³ =r₁₁ = r₁₁
Transcribed Image Text:The problems are generally based on the following model: A particular spacecraft can be represented as a single axisymmetric rigid body B. Let n₂ be inertially fixed unit vectors; then, 6, are parallel to central, principal axes. To make the mathematics simpler, introduce a frame C where n₂ = ĉ₁ = b; initially. 6₁ Assume a mass distribution such that J =₁₁• B* •b₁ = 450 kg - m² I = b² •Ï¾˜ • b₂ = b¸ •Ï¾* •b¸ = 200 kg - m² K J-I C³ =r₁₁ = r₁₁
Assume that the body moves in a circular orbit at a constant rate Q. Let â₁ be orbit-fixed unit vectors where a₂ is directed from the orbit center to B*; then â³ is 90° from a₂ in the direction of motion and â₁ is parallel to the orbit normal. ده * The dependent variables in the differential equations that govern motion include the measure numbers ₁; the kinematic variables that are the body 1-3-2 angles for orientation of C in A: @₁ = 0 w₂ = -Kw₁₂-rw₁₂+3KQ² (c₁₂S¸ − c₁₁ ) (c₁₁₁ + §¸§₁ ) w₁ = Kw₁₂+rw₂ −3KQ² (c₁S₂c₁₂ + $₁₂S₁) c₁₂ Ġ₁ = [(@₁c¸ + @¸§¸ − rc³)/2] - Ω ė₁₂ = − (@₁ − 1 ) s³ + W₂cz - - Ò¸ = {[@¸¢¸ − r¢¸+@₂$¸ ] $₂ / €₂ } + @₂ $2 The motion of interest is a constant spin of B in N about an axis parallel to the orbit normal, i.e., ³ = @â₁ = nÒâ₁, for some given n. Select 7 appropriately to keep C fixed in A during the nominal motion. If r = x2, what is the proper choice of x? Why? What is the resulting particular solution of the differential equations that corresponds to the nominal motion? Demonstrate that it satisfies the differential equations.
Transcribed Image Text:Assume that the body moves in a circular orbit at a constant rate Q. Let â₁ be orbit-fixed unit vectors where a₂ is directed from the orbit center to B*; then â³ is 90° from a₂ in the direction of motion and â₁ is parallel to the orbit normal. ده * The dependent variables in the differential equations that govern motion include the measure numbers ₁; the kinematic variables that are the body 1-3-2 angles for orientation of C in A: @₁ = 0 w₂ = -Kw₁₂-rw₁₂+3KQ² (c₁₂S¸ − c₁₁ ) (c₁₁₁ + §¸§₁ ) w₁ = Kw₁₂+rw₂ −3KQ² (c₁S₂c₁₂ + $₁₂S₁) c₁₂ Ġ₁ = [(@₁c¸ + @¸§¸ − rc³)/2] - Ω ė₁₂ = − (@₁ − 1 ) s³ + W₂cz - - Ò¸ = {[@¸¢¸ − r¢¸+@₂$¸ ] $₂ / €₂ } + @₂ $2 The motion of interest is a constant spin of B in N about an axis parallel to the orbit normal, i.e., ³ = @â₁ = nÒâ₁, for some given n. Select 7 appropriately to keep C fixed in A during the nominal motion. If r = x2, what is the proper choice of x? Why? What is the resulting particular solution of the differential equations that corresponds to the nominal motion? Demonstrate that it satisfies the differential equations.
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